I hereby certify this document contains my

own original work, and all references are

cited and acknowledged.

signed

John Middendorf

Left and above: The Bluesat micro satellite

showing the tray design.

PREFACE

The purpose of this assignment is to look at the peak vibration response of two different sat-

ellite designs using the finite element analysis Patran/Nastran programs. The satellites are

based on actual satellite designs but are greatly simplified, and modeled so as to have identi-

cal dimensions and mass for comparison purposes.

The first satellite, the Bluesat, is a small unpowered communications satellite designed by stu-

dents at the University of New South Wales. The launch date is scheduled in 2004. The main

feature of the BLUEsat microsatellite structure is the configuration of the mainframe structure,

a series of trays referred to as “module stack frames”. The module trays are stacked on top of

one another to form the satellite mainframe structures. This satellite will be modeled as a stack

of four frames bolted together with a single point mass payload of 10kg located at the center of

the satellite on the middle tray. References: Web, http://www.bluesat.unsw.edu.au/

The second satellite is the Forte launched into orbit for the US

Department of Energy in 1997, with the objective is to record

atmospheric bursts of electromagnetic radiation. The Forte was

one of the first all composite satellite structures and was exten-

sively analyzed prior to launch, including its vibrations modes.

The dynamic

response char-

acteristics of the

composite satel-

lite are affected

by its relatively

high stiffness and

its low damping.

Results of the

actual satellite are

shown here.

Above: the Forte satellite

showing the frame structure.

Below: the composite satellite

being tested and analyzed.

Source: COMPOSITE SATELLITE STRUCTURE FOR FORTÉ Los Alamos National Laboratory

(Proceedings from The Tenth International Conference on Composite Materials 1995)

MODELING THE SATELLITES

For the purposes of the assignment, simplified models were created in Patran and analyzed

using Nastran finite element software. Dimensions are based on the last digit of my student

number (1), specifying L=220mm. The BlueSat was modeled as four simple trays bolted to-

gether with a cover plate on the top. All side, floor, and top plates were modeled as 4mm thick,

and the corner posts were modeled as 22mm square beams.

Left: Dimensions. Above: BlueSat simplifications.

For the model of Forte design, a simple frame with alternating cross

sectional diagonal braces structure replaced the tray design. The

frame cross section spec was given as 50mm square, with a thick-

ness determined by equalizing the mass of the structure to be identi-

cal to the Bluesat model. The model also included a 4mm thick floor

panel on the bottom of each of the frame trays. Note that the 50mm

square beam is an unlikely size in conjunction with the specified

governing dimension(L=220mm), as the trays are only 55mm high(L/

4). However, for modeling the idea is to look at how a square tubular

reinforced deck improves the frequency response.

MATERIAL:

Material for both satellites was modeled as aluminum with a tensile strength of 71

GPa, a Poisson ratio of 0.334, and a density of 2710 kg/m3.

DAMPING COEFFICIENT:

Structural damping ratios for the structures was input within the

material property section as 0.07 based on Robert Cook’s recommendation for bolted struc-

tures (p. 234).

UNITS:

Throughout the analysis care was taken to enter a consistent unit system, as the pro-

grams uses dimensionless values. Input values are scaled from a normal N*kG*m*s system

into a N*Mg*mm*s system.

Dimension

Unit

Example =

Units input in Patran

Length

mm

(10

-3

m)

210mm

210

Mass

Mg (10

6

grams)

15 kg(modeled weight of satellite w/ 10kg payload) 15 e-3 (0.015)

Force

Mg mm/s

2

(N)

4 g load on 15 e3 Mg (4 x 9810 mm/s

2

x15 e3 Mg) 5.89 e8

Pressure

Mg /mm s

2

(10

6

Pa) 1 Pascal (N/m

2

)

1 e-6

Density

Mg/mm

3

(10

12

kg/m

3

) 2710 kg/m

3

(density of Aluminum)

2.71 e-9

Elastic Modulus Mg /mm s

2

(10

6

Pa) 71 GPa

71 e3 (71000)

Above: two of the four

“frame trays” for modeling

the Forte satellite.

L

L/4

L/10

MODELING STRATEGY

General Procedure

Using Patran, extensive use of groups was utilized to aid in the modeling of the satellites. A coarse mesh

was initially created and an initial analysis run, then the mesh was refined and more extensive analyses were

performed. The general procedure was thus:

1. Create the

Geometry

of a single tray. Groups were created based on the geometry and properties. For

the BlueSat, 3 groups were created for the tray design: beams, floor plate, and side plates. For the Forte, 4

groups were created: horizontal beams, vertical beams, diagonal beams, and floor plate. Once the groups for

a single tray were created, it was a simple matter to transform each group to create additional trays, though

some care was required in the case of the Forte design so as to not duplicate structural members.

2. Next, enter

Materials

and

Properties

of the structure. The material throughout was aluminum (proper-

ties listed previous page, including damping coefficient of 0.07). Since the properties were all aligned with

the geometry, properties were assigned to the geometry which allowed ease of refining the mesh (although,

for safe measure, the properties were reapplied to each group after each mesh creation/refinement). Beams

were modeled as 1D Beams using sections from the Patran beam library (22mm solid square for the BlueSat,

50mm hollow square for the Forte), with orientation vectors aligned perpendicular to the base. Each group of

beams (with different orientations) were assigned a unique property set with the orientation vector aligned ap-

propriately. Plates were modeled as 2D Shells, with a thickness of 4mm.

3. Create

Elements

. Two meshes were created for each design: a coarse mesh and a fine mesh (note:

normal mode and frequency response results were nearly identical for both meshes in each design, which

indicated that an additional finer mesh was not required). The general procedure was to create mesh seeds

on each group, then mesh the elements, prior to mesh seeding and meshing the next group. This ensured

consistent and deliberate meshing results. Curves were modeled as bar2 elements, and surfaces as quad4

isomeshes. More advanced elements were not considered necessary for this modal and frequency response

analysis. Initial coarse meshes had an element length of 20, while the fine meshes had an element length of

10. Finally, all groups were posted and Equivalence task was performed, eliminating many duplicate nodes.

4. Add the Lumped payload mass. The first task was to check the mass of the entire structure and to check

for realistic accuracy. This was done using the Tools>Mass Properties command in Patran. (results=5.6e-3)

Then, using

Elements

, a point element was created at the center of the middle tray. Then, in

Properties

, a

0D lumped payload mass of 10KG was created and assigned to the point element in the center of the struc-

ture. Again total mass was checked (results= 1.56e-2) This completed the structural modeling of each satel-

lite.

5.Next, in

Loads/BC

, fix the bottom four corners in the x and y directions using a translation vector input of

<0,0, >. Actually, tests were run without this boundary condition, but as we were mainly interested in the ver-

tical frequency response of the structure, fixing the bottom four corners of the satellite in the x and y directions

allowed for a more relevant result set.

6. In

Analysis

, run a Normal Modes analysis which gives the natural frequencies of vibration (harmonic fre-

quencies) of the structure. Check normal modes and frequencies using a quickplot of translational eigenvec-

tors in

Results.

7. Add the 4g vertical force. In

Elements

, create a point element at the center of the bottom tray. In

Proper-

ties

, add a 0D lumped point mass that is 106 times the mass of the structure (1.5e4). In

Loads/BC

, add the

vertical force that corresponds to 4g (<0 0 5.886 e8>). This force is automatically multiplied by a timewise

sine function in the frequency response analysis.

8. Finally, in

Analysis

, run a Frequency Response analysis, create XY plots in

Results

of displacement

verses frequency, and refine results further with specific subcase parameter frequency ranges depending on

the range of interest.

MODELS

Above: BlueSat beam and plate structure-

-single tray (with coarse mesh). Note the

center point in the floor plate which was

considered necessary for the point ele-

ments for the loads.

Above: Forte beam structure . Points were

created, then linked with curves. Groups

were used to produce successive layers.

Above: BlueSat 4 tray structure. (fine mesh). Beams

shown in 3d view. Note the use of Groups on right.

After some consideration, the side panels were mod-

eled as bonded (nodes were equivalenced), Initial tests

indicated that the frequency response of floor plates

were relatively independent of side plate bonding. The

model has a bonded panel on top which is based on the

actual design.

Above: Forte completed model. Shown in 1D view of beams.

Note the use of groups on the right which were chosen based

on the property values required for each individual orientation

of beam direction. Floors are 4mm plate structures. The top

lacks a plate structure as this more accurately represented the

actual design.

Above: Initial test of single tray normal modes, and

modal shape for first harmonic frequency (196 Hz).

Above: 3D full span view of Forte beam structure. Note that

the 50mm specified beam dimension nearly completely fills

in the 220mm x 220mm x 220 mm structure. Joint connec-

tions are default. This prompted some thought as to how

Nastran actually analyzes beam connections. Since the area

of frequency response interest in the frequency response of

the floor plates, the line modeling of beams was considered

adequate.

RESULTS

Tests were fun for both the fine and coarse mesh versions of the satellite structures. The re-

sults were nearly identical, as shown below for the BlueSat coarse and fine mesh. Forte coarse

and fine mesh runs were also nearly identical. Furthermore, the Nastran recommends 5 to 10

grid points per half cycle of response amplitude, both meshes are within these limits. The fine

mesh results are presented hereafter.

ABOVE: Coarse mesh Normal Mode results and Frequency

Response XY displacement vs. frequency plot. First har-

monic frequency at 37.7 Hz,second harmonic at 120.0Hz

(note: slightly different boundary condition applied)

ABOVE: Coarse mesh Normal Mode results and Frequency

Response XY plot (displacement vs. frequency). First har-

monic frequency at 37.8Hz, second at 120.6Hz. Note almost

identical results of frequency response from coarse mesh at

left.

BlueSat Normal Modes

Above: BlueSat Normal Modes results. Fringe plot of first harmonic frequency. The sideplate and beam groups have been

unposted for clarity. The BlueSat has a low first harmonic frequency of 37.8 Hz, and a second harmonic frequency at 120Hz.

At the first harmonic frequency, we see a modal shape that includes displacement of all the floor plates, with larger displace-

ments of the two floorplates at the point at which the point masses were added. Note the opposing displacements of the two

point masses. This image represents the moment when the force on the larger mass is at a minimum (-4g), while the payload

mass's inertia is responding accordingly.

BlueSat Frequency Response

Left: The Frequency Response XY plot

of displacements vs. frequency for the

BlueSat satellite. We see a large peak at

around 37 Hz of around 12mm displace-

ment. The unsupported 4mm thick floor

plate in resonance will displace 11 mm in

both directions under the sinusoidal load

condition. This is a large displacement

and would be unsatisfactory if delicate

instruments were installed in the satel-

lite. However, in reality, the BlueSat is

an unpowered communications satellite

and is unlikely to receive a force input

that would approach its natural harmonic

frequency.

Left: a close up of the region of inter-

est with 100 divisions of frequency

between the ranges of 25Hz and 50Hz.

Here we can find a more exact value

of the displacement. From the chart,

we find the maximum displacement of

11.2 mm at 37 Hz.

Above: XY plot in the region of the second harmonic frequency and modal shape at the second harmonic frequency.

The plot gives results of 0.11mm at 120Hz. Note that in the second harmonic frequency the lower mass and middle

mass are in phase with eachother (at the first harmonic, they were 180 degrees out of phase with eachother).

Summary of BlueSat

Displacement Results

First Frequency = 37.8 Hz

Max. displacement = 11.2 mm

Second Frequency = 120 Hz

Max displacement = 0.11mm

BlueSat Large Mass Force Response

Forte Analysis

Before launching into the Forte analysis proper, and with the Bluesat results still in mind, it is

interesting to look at the Forte design without the diagonal cross braces. Results are similar

to the Bluesat design, indicating that the differences of the Forte design are largely due to the

diagonal cross braces below the floor plate.Below is the result from a run of the Forte Satellite

design without the diagonal cross braces underneath the floor plate. Normal modes are similar

to the BlueSat design (first harmonic frequency at 33Hz, and second harmonic frequency at

105Hz). Displacements are likewise similar to the BlueSat design. What this indicates is that

the weight savings from using a tubular structure over a plate design can be applied to adding

floor reinforcements to improve the frequency response of a design.

Above: XY plots of the Large Mass displacement response from the 4g force. The first chart (left) shows a gradual decline

in displacement, as would be expected: as the frequency of the applied load increases, the force approaches a steady value

and the inertia of the mass is not accelerated. Right: close up of displacement in the region of the first harmonic frequency.

Displacement at 37 Hz=0.75mm. This allows us to check the input values, below.

Check of the force input

Theoretically, with no damping, once motion on a mass begins as a result of a force, the

mass vibrates with simple harmonic motion, with the instantaneous

displacement=amplitude * sin(wt), where w (omega) is given in radians.

Differentiating, we get:

acceleration = w

2

*amplitude* sin(wt), which is at a maximum when sin(wt) =1.

Since we know the acceleration (4g), we solve: 4*9810=0.75*(37*2*pi)

2,

which solves:

39240=40534, an error of 3.3%.

Left: Forte frame design

results with the diagonal

cross braces under the

floor plates removed.

Forte Results: Normal Modes

Above: Forte Normal Modes results. Fringe plot of first harmonic frequency. The Forte has a higher first harmon-

ic frequency of 156.9 Hz, and a second harmonic frequency at 534.1Hz. At the first harmonic frequency, we see a

modal shape that includes displacement of all the floor plates, with larger displacements of the two floorplates at

the point at which the point masses were added. Again we see 180 degree out-of-phase opposing displacements

of the two point masses. This image represents the moment when the force on the larger mass is at a maximum

(4g), while the payload mass's inertia is responding accordingly.

Normal Mode Result Discussion

Both satellites have similar modal deformations, but the BlueSat first and second harmonic

frequencies (37.8Hz and 120Hz) are much lower than for the Forte design (157Hz and

534Hz). This is expected as the tubular beam structure is much more rigid than the plate

structure. The floor plates are modeled identically, but the Forte has the addition of two 50mm

square tubular beams supporting the floor. This clearly stiffens the structure and increases

the frequencies of the normal modes.

Tubular Beams

In order to equalize the masses of the two satellites (5.6KG), the square tubular beam thick-

ness was set at 0.66mm, very thin for such a large tubular beam. Further analysis would indi-

cate investigating buckling modes of this structure, but this analysis did not encompass stress

analysis.

Forte Frequency Response

Left: Frequency response of the Forte

Satellite. As predicted by the normal mode

analysis, we see a peak at around 150 Hz,

with a smaller bump somewhere around 550

Hz. Displacements are much lower than

the BlueSat design, at around 0.7mm. The

curve does not have a static value at zero

frequency, as it should. This is probably

because the satellite is not constrained by

fixed boundary conditions and the software

plots the calculated zero frequency displace-

ments (rigid body modes), which physically

relate to the unbounded travel of the satellite

structures under a given force.

Left: Close up of the displacement/

frequency XY plot in the region of interest

of the first harmonic frequency, from 125Hz

to 175Hz. We see a peak at 156Hz and a

maximum displacement of 0.64mm, a small

value in comparison with the 11.2mm figure

for the BlueSat design. This value for maxi-

mum vibrational displacement is probably

adequate for sensitive electronic equipment

on board a typical satellite.

Left: Displacement/Frequency XY plot in

the region of the second harmonic frequency

(475Hz to 575Hz). Here we see a minimal

displacement (0.006mm) at the second har-

monic (534Hz).

Summary of Forte

Displacement Results

First Frequency = 156 Hz

Max. displacement = 0.64 mm

Second Frequency = 534 Hz

Max displacement = 0.006 mm

Forte large mass force response

Above: XY plots of the Large Mass displacement response from the 4g force in the Forte satellite. The first chart (left)

shows a gradual decline in displacement, as would be expected: as the frequency of the applied load increases, the force

approaches a steady value and the inertia of the mass is not accelerated. Right: close up of displacement in the region of the

first harmonic frequency. Displacement at 156Hz=0.041mm. This allows us to check the input values, below.

Check of the force input

Theoretically, with no damping, once motion on a mass begins as a result of a force, the

mass vibrates with simple harmonic motion, with the instantaneous

displacement=amplitude * sin(wt), where w (omega) is given in radians.

Differentiating, we get:

acceleration = w

2

*amplitude* sin(wt), which is at a maximum when sin(wt) =1.

Since we know the acceleration (4g), we solve: 4*9810=0.041*(156*2*pi)

2,

which solves:

39240=39391, an error of 0.4%.

Summary

Final results show a difference by a factor of 17.5 between the maximum vibrational displace-

ment of the Bluesat design (11.2mm at 37Hz) and the Forte frame design (0.64mm at 157Hz).

Much of the discussion of the results is included in the captions of the images already pre-

sented above. To summarize, although many assumptions were made in modeling the two

satellite designs, a frame structure of the same given mass of a plate structure offers more

rigidity, has higher natural frequencies, and will displace less at the maximum resonance. As-

sumptions included the way the structure was modeled (rigid corners), damping coefficient,

mass considerations, and boundary conditions. The models used a damping coefficient of

0.07, which is recommended by Cook for bolted structures. The damping coefficient is the

ratio of the amount of damping as a percentage of critical damping (no oscillation) and influ-

ences the maximum displacement for a given vibrational frequency (if damping =0, displace-

ment of a forced response goes to infinity). Clearly, the damping value chosen affects the

values presented here. The payload mass was modeled as a point mass, which also could

affect the vibrational modes. A distributed load could offer more accurate results. In regard

to boundary conditions, the model was underconstrained as is allowed in vibration analyses

though the bottom corners were fixed in the x and y directions. Unfixing the corners gave

slightly different results in regards to the harmonic frequencies (but not in the critical regime

in the first and second natural frequencies). In summary, the models presented here give a

good indication of two approaches to satellite design and the Nastran/Patran finite element

analysis gives viable results for determining modal shapes, harmonic frequencies, and forced

vibration responses of a structure.

References: Cook, Robert D. Finite Element Modeling for Stress Analysis. John Wiley and Sons, 1995.

Nastran/Patran Manuals.